Absolute points of correlations of $PG(3,q^n)$

https://doi.org/10.1007/s10801-020-00970-3

Absolute points of correlations of

PG(3, q

)

Received: 14 November 2019 / Accepted: 4 August 2020 © The Author(s) 2020

Abstract

The sets of the absolute points of (possibly degenerate) polarities of a projective space are well known. The sets of the absolute points of (possibly degenerate) correlations, different from polarities, of PG(2, qn_{), have been completely determined by B.C.}

Kestenband in 11 papers from 1990 to 2014, for non-degenerate correlations and by D’haeseleer and Durante (Electron J Combin 27(2):2–32, 2020) for degenerate correlations. In this paper, we completely determine the sets of the absolute points of degenerate correlations, different from degenerate polarities, of a projective space PG(3, qn). As an application we show that, for q even, some of these sets are related to the Segre’s(2h+ 1)-arc of PG(3, 2n) and to the Lüneburg spread of PG(3, 22h+1).

Keywords Sesquilinear forms· Correlations · Polarities Mathematics Subject Classification 51E20· 05B25

1 Introduction and preliminary results

Let V and W be two F-vector spaces, where F is a field. A map f : V −→ W is called semilinear orσ-linear if there exists an automorphism σ of F such that

f(v + v) = f (v) + f (v) and f (av) = aσ f(v)

for all vectorsv ∈ V and all scalars a ∈ F.

Ifσ is the identity map, then f is a usual linear map.

B

1 _{Dipartimento di Matematica e Applicazioni “Caccioppoli”, Università di Napoli “Federico II”,}

Let V be an F-vector space with finite dimension d. A map  ,  : (v, v) ∈ V × V −→ v, v ∈ F

is a sesquilinear form or a semibilinear form on V if it is a linear map on the first argument and it is aσ-linear map on the second argument, that is:

v + v, v = v, v + v, v, v, v+ v = v, v + v, v,

av, v = av, v, v, av = aσv, v,

for allv, v, v∈ V , a ∈ F and σ an automorphism of F. If σ is the identity map, then  ,  is an usual bilinear form.

If B = (e1, e2, . . . , ed+1) is an ordered basis of V , then for x, y ∈ V we have

x, y = Xt_AYσ_{, where A= (e}

i, ej) is the associated matrix to the sesquilinear

form with respect to the ordered basisB, X and Y are the columns of the coordinates of x, y w.r.t. B. The term sequi comes from the Latin, and it means one and a half. For every subspace S of V , put

S:= {y ∈ V : x, y = 0 ∀x ∈ S}, S⊥:= {y ∈ V : y, x = 0 ∀x ∈ S}.

Both Sand S⊥are subspaces of V . The subspaces Vand V⊥are called the right and the left radical of ,  and will be also denoted by Radr(V ) and Radl(V ),

respectively.

Proposition 1.1 The right and the left radicals of a sesquilinear form of a vector space V have the same dimension.

Proof Indeed, dimV _{= d + 1 − rk(B), where B = (a}σ−1

i j ) and dimV⊥ = d +

1− rk(At), where At is the transposed of the matrix A. The assertion follows from

rk(B) = rk(A) = rk(At). 

A non-degenerate sesquilinear form ,  has V⊥= V= {0}.

Definition 1 Aσ -sesquilinear form is reflexive if ∀ u, v ∈ V it is:

u, v = 0 ⇐⇒ v, u = 0.

Definition 2 Let V be an F-vector space of dimension greater than two. A bijection g : PG(V ) −→ PG(V ) is a collineation if g, together with g−1, maps lines into lines. If

V has dimension two, then a collineation is a mapv ∈ PG(V ) −→  f (v) ∈ PG(V ),

induced by a bijective semilinear map f : V −→ V .

Theorem 1.2 (Fundamental Theorem) Let V be an F-vector space. Every collineation of PG(V ) is induced by a bijective semilinear map f : V −→ V .

In the sequel if S is a vector subspace of V , we will denote with the same symbol S the associated projective subspace of PG(V ).

Definition 3 Let f : V −→ V be a semilinear map, with K er f = {0}. The map

v ∈ PG(V )\K er f −→  f (v) ∈ PG(V ), will be called a degenerate collineation of PG(V ).

Definition 4 A (degenerate) correlation or (degenerate) duality of PG(d, F) is a

(degenerate) collineation between PG(d, F) and its dual space PG(d, F)∗.

Remark 1.3 A correlation of PG(d, F) can be seen as a bijective map of PG(d, F)

that maps k-dimensional subspaces into(d − 1 − k)-dimensional subspaces reversing inclusion and preserving incidence.

A correlation of PG(d, F) applied twice gives a collineation of PG(d, F).

Theorem 1.4 Any (possibly degenerate) correlation of PG(d, F), d > 1, is induced by aσ -sesquilinear form of the underlying vector space Fd+1. Conversely, every σ-sesquilinear form of Fd+1induces two (possibly degenerate) correlations of PG(d, F). The two correlations agree if and only if the form ,  is reflexive.

Proof A (possibly degenerate) duality of PG(d, F), d > 1, is induced by a σ-linear

transformation f of Fd+1into its dual, since it is a (possibly degenerate) collineation. Define a map ,  : Fd+1× Fd+1−→ F in the following way:

u, v = f (v)(u),

that is the result of applying the element f(v) of (Fd+1)∗to u. It follows that ,  is aσ -sesquilinear form. Indeed, it is easy to see that  ,  is linear on the first argument and since

u, v1+ v1 = f (v1+ v2)(u) = f (v1)(u) + f (v2)(u) = u, v1 + u, v2,

and

u, av = f (av)(u) = aσ f(v)(u) = aσu, v,

it is semilinear on the second argument. Moreover, ,  is non-degenerate if and only if f is a bijection.

Conversely, anyσ-sesquilinear form on V = Fd+1induces two (possibly degenerate) correlations of PG(d, F) given by

⊥: P = u ∈ PG(d, F)\Radl(V ) → P⊥= {v | u, v = 0} ∈ PG(d, F)∗

and

Remark 1.5 A (possibly degenerate) correlation induced by a σ-sesquilinear form will

be also called aσ -correlation. Sometimes we will call linear a (degenerate) correlation whose associated form is a bilinear form.

1.2 Reflexive sesquilinear forms and polarities

Definition 5 A (degenerate) polarity is a (degenerate) correlation whose square is the

identity.

If⊥ is a (possibly degenerate) polarity, then for every pair of points P and R the following holds:

P ∈ R⊥ ⇐⇒ R ∈ P⊥.

Proposition 1.6 A (degenerate) correlation is a (degenerate) polarity if and only if the induced sesquilinear form is reflexive.

Proof Let  ,  be a reflexive σ-sesquilinear form of Fd₊₁

q . If u∈ v⊥, thenv ∈ u⊥.

Hence, the mapu → u⊥ defines a (possibly degenerate) polarity. Conversely, given a (possibly degenerate) polarity⊥, if v ∈ u⊥, then u∈ v⊥. So, the induced

sesquilinear form is reflexive. 

The non-degenerate, reflexiveσ-sesquilinear forms of a (d +1)-dimensional F-vector space V have been classified (for a proof see, e.g., Theorem 3.6 in [1] or Theorem 6.3 and Proposition 6.4 in [5]).

Theorem 1.7 Let ,  be a non-degenerate, reflexive σ-sesquilinear form of a (d +1)-dimensional F-vector space. Then, up to a scalar factor, the form ,  is one of the following:

(i) a symmetric form, i.e.,

∀ u, v ∈ V u, v = v, u (char(F) = 2 ⇒ ∃ v ∈ V : v, v = 0), (ii) an alternating form, i.e.,

∀ v ∈ V v, v = 0 (d is necessarily odd), (iii) a Hermitian form, i.e.,

∀ u, v ∈ V u, v = v, uσ (σ2_{= 1, σ = 1).}

Let V be a(d + 1)-dimensional F-vector space and consider the associated projective geometry PG(d, F). If V is equipped with a sesquilinear from  , , we may consider in PG(d, F) the set  of absolute points of the associated correlations, that is the points

X such that X ∈ X⊥(or equivalently X ∈ X). If A is the associated matrix to the

σ -sesquilinear form  ,  w.r.t. an ordered basis of V , then the set  has equation XtA Xσ = 0.

From the previous theorem, the polarities of PG(d, F) are in one to one correspon-dence with non-degenerate, reflexiveσ-sesquilinear forms of Fd+1. Hence, to every polarity of PG(d, F) there is an associated pair (A, σ), with A a non-singular matrix of order d + 1 and σ an automorphism of F. In what follows, we will identify a polarity with a pair(A, σ). From the last theorem of the previous section, we have the following:

Theorem 1.8 Let(A, σ) be a polarity of PG(d, q), one of the following holds:

(i) σ = 1, A is a symmetric matrix. The polarity is called an orthogonal polarity. If

q is even, there is a non-absolute point.

(ii) σ = 1, A is a skew-symmetric matrix, d is odd. Every point is an absolute point

and the polarity is called a symplectic polarity.

(iii) σ2= 1, so σ : x → x√q, q is a square, A is a Hermitian matrix. The polarity

is called a Hermitian or unitary polarity.

Recall that a square matrix A is either symmetric if A = At, or skew-symmetric if A= −At or Hermitian if A = Aσt, σ2= 1, σ = 1. Each of the above polarities of

PG(d, q) determines a set  : XtA Xσ = 0, as set of its absolute points. The three

types of polarities give rise to the following, well known, subsets of PG(d, q):

Definition 6 If(A, σ) is a polarity of PG(d, q), then one of the following holds:

(i)  is called a quadric of PG(d, q) (q odd, orthogonal polarity).  is a hyperplane of PG(d, q) (q even, orthogonal polarity).

(ii)  is the full pointset of PG(d, q) (d odd, symplectic polarity) and the geometry determined is a symplectic polar space.

(iii)  is a Hermitian variety of PG(d, q) (q a square, σ2 = 1, σ = 1, unitary polarity).

Remark 1.9 If q is even, σ = 1 and (A, σ) is an orthogonal polarity, so A is a symmetric

matrix, then the set of its absolute points is a hyperplane. Note that in many books (but not in the book of P. Dembowski [10]) this kind of polarity is called a pseudo

polarity.

In all the cases of the previous definition, the set is often called also non-degenerate since the associatedσ -sesquilinear form is non-degenerate. All the above sets have been classified, and for each of them it is possible to give a canonical equation (see, e.g., chapters 22 and 23 in [16]).

Theorem 1.10 If(A, σ) is a polarity of PG(d, q), then let  : XtA Xσ = 0 be the set of its absolute points. The following hold:

(i) If is a quadric, so q is odd, then we have: (1) If d is even, then

(2) If d is odd, then either

 = Q−_{(d, q) : x}

1x2+ · · · + αxd2+ βxdxd+1+ γ xd2+1= 0, with αx2

d+βxdxd+1+γ xd2+1irreducible polynomial overFq(elliptic quadric) or  = Q+(d, q) : x1x2+ · · · + xdxd₊₁= 0 (hyperbolic quadric).

If is a hyperplane, so q is even, then  : x1= 0.

(ii) If is a symplectic polar space, then  is the full pointset of PG(d, q), d odd.

The geometry determined will be denoted by W(d, q). A canonical form for the associated bilinear form is

x, y = x1y2− x2y1+ x3y4− x4y3+ · · · + xdyd+1− xd+1yd.

(iii) If is a Hermitian variety, then

 = H(d, q) : x1√q+1+ x √_q+1

2 + · · · + x √_q+1 d+1 = 0.

Remark 1.11 Regarding degenerate, reflexive σ-sesquilinear forms of V = Fd+1 q , it

is possible to prove that if dim V⊥= r + 1, r ≥ 0, then the set  of absolute points in PG(d, q) of the associated degenerate polarity is, in all the possible cases, a cone

(Vr, Qd−1−r) with vertex a subspace Vr, of dimension r , projecting the set Qd−1−r

of absolute points of a polarity in a subspace Sd_−1−r, of dimension d−1−r, skew with Vr. The set Qd_−1−rcan be either a quadric or a Hermitian variety (if q is a square) or

the full pointset of PG(d −1−r, q) (a symplectic geometry). In these cases, we call the set of the absolute points either a degenerate quadric,or a degenerate Hermitian variety or a degenerate symplectic geometry, respectively. The knowledge of the set of the absolute points of a polarity of Sd−1−rdetermines also the knowledge of the set of the

absolute points of a degenerate polarity. If Qd−1−ris the full pointset of PG(d −1−r),

then d and r must have the same parity and the cone(Vr, PG(d − 1 − r, q)) is a quotient geometry PG(d, q)/Vr.

Sometimes we will call non-degenerate quadrics, non-degenerate symplectic geome-tries and non-degenerate Hermitian varieties, the set of the absolute points of a polarity. The following proposition characterizes these sets.

Proposition 1.12 Let be either a non-degenerate quadric or a non-degenerate sym-plectic polar space or a non-degenerate Hermitian variety of PG(d, q) and denote by

⊥ the associated polarity. For any point Y of , the set Y⊥_{is a hyperplane meeting}  in a cone (Y , Qd−2), where Qd−2is either a degenerate quadric or a non-degenerate symplectic polar space or a non-non-degenerate Hermitian variety, respectively and vice versa.

1.3 Sets of the absolute points of a linear correlation of PG(d, F)

Let be the set of the absolute points of a linear correlation of PG(d, F) with equation

XtA X = 0, |A| = 0. Assuming that  is a proper subset of PG(d, F), the matrix A

cannot be a skew-symmetric matrix and is a (possibly degenerate) quadric and vice versa. Note that the previous holds, independently of the characteristic of the field F. If the characteristic of F is 2, then there is no relation between|A| and degeneracy or not of the quadric. Observe that, not assuming that A is a symmetric matrix, also for characteristic of F either odd or 0, this relation is lost and, a bit surprisingly, the following holds:

Proposition 1.13 If : XtA X = 0 is a proper subset of points of PG(d, F), with

|A| = 0, then  is a (possibly degenerate) quadric. Vice versa, if  is a non-empty

(possibly degenerate) quadric of PG(d, F), then there exists a matrix A, with |A| = 0, s.t. is projectively equivalent to the set of points with equation XtA X= 0.

Moreover, since in the finite Desarguesian projective spaces all the quadrics are non-empty (see, e.g., [38]), we have the following:

Proposition 1.14 If : XtA X = 0, with |A| = 0, is a set of points of PG(d, qn), then is either a degenerate symplectic geometry or a (possibly degenerate) quadric. Vice versa if is either a degenerate symplectic geometry or a (possibly degenerate) quadric of PG(d, qn_{), then there is a matrix A with |A| = 0 s.t.  is projectively} equivalent to the set of points with equation XtA X = 0.

1.4FFFqqq-linear sets of PG(d, qn)

Definition 7 Let = PG(r − 1, qn_{), q = p}h_{, p a prime. A set L is said anF} q-linear set of of rank t if it is defined by the nonzero vectors of an Fq-vector subspace U of V = Fr_qn of dimension t, that is

L = LU = {uqn : u ∈ U\{0}}.

Let = PG(W, Fqn) be a subspace of  of dimension s, we say that  has weight i with respect to LU if dimFq(W ∩ U) = i. An Fq-linear set LU of of rank t is said to be scattered if each point of LU has weight 1, with respect to LU.

In [2] it is proved that if LU is a scatteredFq linear set of , then t ≤ rn/2. It

is clear, by definition that LU is a scattered Fq-linear set of rank t if and only if

|LU| = qt−1+ qt−2+ · · · + q + 1. If L is a scattered linear set of PG(r − 1, qn) of

rank r n/2, it is called a maximum scattered linear set.

If dimFqU = dimFqnV = r and UFqn = V , then LU ∼= PG(U, Fq) is a subgeometry of. In such a case, each point has weight 1, and hence |LU| = qr−1+ qr−2+ · · · +

q+ 1. Let = PG(t, q) be a subgeometry of = PG(t, qn_{) and suppose that there}

exists a(t −r)-dimensional subspace of disjoint from . Let  = PG(r −1, qn₎

of from to, i.e.,  = {, x ∩  : x ∈ }. Let p__,be the map from \ to defined by x → , x ∩ . We call the center and the axis of p__,. In [31] the following characterization ofFq-linear sets is given:

Theorem 1.15 If L is a projection of PG(t, q) into  = PG(r − 1, qn_{), then L is an}

Fq-linear set of of rank t + 1 and L = . Conversely, if L is an Fq-linear set of of rank t+ 1 and L = , then either L is a subgeometry of  or there are a (t − r)-dimensional subspaceof = PG(t, qn) disjoint from  and a subgeometry of _{disjoint from}_{such that L= p,( ).}

A family of maximum scattered linear sets to which a geometric structure, called pseudoregulus, can be associated has been defined in [30].

Definition 8 Let L= LU be a scatteredFq-linear set of = PG(2n − 1, qt) of rank t n, t, n ≥ 2. We say that L is of pseudoregulus type if:

(i) there exist m = q_qntt₋₁−1pairwise disjoint lines of, say s1, s2, . . . , sm, such that wL(si) = t, i.e., |L ∩ si| = qt−1_{+ q}t−2_{+ · · · + q + 1 ∀ i = 1, . . . , m,}

(ii) there exist exactly two (n − 1)-dimensional subspaces T1and T2of disjoint

from L such that Tj∩ si = ∅ ∀ i = 1, . . . , m and j = 1, 2.

We call the set of linesPL = {si : i = 1, . . . , m} the Fq-pseudoregulus (or simply the pseudoregulus) of associated with L and we refer to T1and T2as transversal spaces

ofPL (or transversal spaces of L). When t= n = 2, these objects already appeared first in [15], where the term pseudoregulus was introduced for the first time. See also [11].

In [12] and in [30] the following class of maximum scatteredFq-linear sets of the

projective line = PG(1, qt) with a structure resembling that of an Fq-linear set of PG(2n − 1, qt), n, t ≥ 2, of pseudoregulus type has been studied. Let P1 = wqt and P2= vqt be two distinct points of and let τ be an automorphism of Fqt such that Fi x(τ) = Fq. For eachρ ∈ F∗_qt, the set Wρ,τ = {λw + ρλτv : λ ∈ Fqt}, is anFq-vector subspace of V = F2_qt of dimension t and Lρ,τ = LW_ρ,τ is a scattered

Fq-linear set of.

Definition 9 In [30] the linear sets L_ρ,τhave been called of pseudoregulus type and the points P1and P2the transversal points of Lρ,τ. If Lρ,τ∩Lρ,τ = ∅, then Lρ,τ = Lρ,τ. Remark 1.16 (See [12,30]) Note that L_ρ,τ = L_ρ_,τif and only if N(ρ) = N(ρ) (where

N denotes the Norm ofFqt overF_q); so P₁, P₂and the automorphismτ define a set of

q−1 mutually disjoint maximum scattered linear sets of pseudoregulus type admitting

the same transversal points. Such maximal scattered linear sets, together with P1and P2, cover the pointset of PG(1, qt). All the Fq-linear sets of pseudoregulus type in  = PG(1, qt_{), t ≥ 2, are equivalent to the linear set L}

1,σ1 under the action of the

collineation group PL(2, qt).

Remark 1.17 We observe that the pseudoregulus of PG(1, qn_{) is the same as the Norm}

surface(or sphere) of R.H. Bruck, introduced and studied, in the seventies, by R.H. Bruck (see [3,4]) in the setting of circle geometries.

2 The

-quadrics of PG(d, q

)

In this section, we will introduceσ-quadrics of PG(d, qn) and we will determine their intersection with subspaces.

Definition 10 Aσ-quadric of PG(d, qn) is the set of the absolute points of a (possibly

degenerate)σ-correlation, σ = 1, of PG(d, qn). A σ -quadric of PG(2, qn) will be called aσ -conic.

Remark 2.1 Let  : Xt_{A X}σ _{= 0 be the set of the absolute points of a (possibly}

degenerate)σ-correlation of PG(d, q2) and let σ : x → xq. In this case, the set is a (possibly degenerate) Hermitian variety of PG(d, q2) if A is a Hermitian matrix. We have included Hermitian varieties in the definition of a σ -quadric in order to have no exceptions everywhere. Indeed, we will see that Hermitian varieties appear as intersection ofσ-quadrics of PG(d, q2_{) with subspaces.}

Proposition 2.2 Let : XtA Xσ = 0 be a σ-quadric of PG(d, qn). Every subspace S intersects either in a σ-quadric of S or it is contained in .

Proof Let S be an h-dimensional subspace of PG(d, qn_{). We may assume, w.l.o.g.}

that S: xh+2= 0, . . . , xd+1= 0. Let Abe the submatrix of A obtained by deleting

the last d− h rows and columns of A. If A= 0, then S ⊂ . If A = 0, then S ∩ 

is aσ -quadric of S. 

For the remaining part of this chapter, we can assumeσ = 1. Let V = Fd_q+1n , let ,  be a degenerateσ -sesquilinear form with associated (degenerate) correlations ⊥,  and let : XtA Xσ = 0 be the associated σ -quadric. We will denote by L = V⊥and

R= V, the left and right radicals of ,  respectively, seen as subspaces of PG(d, qn) that will be called the vertices of. Before giving the definition of a non-degenerate

σ -quadric, we prove the following:

Proposition 2.3 Let : XtA Xσ = 0 be a σ -quadric of PG(d, qn).

• For every point Y ∈ \R, the hyperplane Y_{is union of lines through Y either} contained or 1-secant or 2-secant to.

• For every point Y ∈ \L, the hyperplane Y⊥_{is union of lines through Y either} contained or 1-secant or 2-secant to.

Proof Let Y ∈ \R and let Z be a point of Y_{. The line Y Z has equations: X =} λY +μZ, (λ, μ) ∈ PG(1, qn_{); hence, Y}_{∩ is determined by the solutions in (λ, μ)}

of the following equation:

YtAYσλσ+1+ YtA Zσλμσ+ ZtAYσλσμ + ZtA Zσμσ+1= 0. (1) In the previous equation, it is YtAYσ = 0, since Y ∈  and ZtAYσ = 0, since Z ∈ Y. Hence, Equation (1) becomes YtA Zσλ + ZtA Zσμ = 0 and four different

cases occur

• If Yt_{A Zσ = 0, Z}t_{A Zσ = 0, then the line Y Z intersects  exactly at the point Y ;}

• If Yt_{A Zσ = 0, Z}t_{A Zσ = 0, then the line Y Z intersects  exactly at the point Y ;}

• If Yt_{A Z}σ _{= 0, Z}t_{A Z}σ _{= 0, then the line Y Z intersects  at Y and at another}

point.

The second part of the statement follows in a similar way. 

Corollary 2.4 Let : Xt_{A X}σ _{= 0 be a σ -quadric of PG(d, q}n_{) and let L = V}⊥_, R= V.

• For every point Y ∈ L, the set Y_{∩  is union of lines through Y .}

• For every point Y ∈ R, the set Y⊥_{∩  is union of lines through Y .}

Remark 2.5 Let  : Xt_{A X}σ _{= 0 be a σ -quadric of PG(d, q}n_{) and let L = V}⊥_and R = Vbe its vertices. For every point Y ∈ \R, Yis the tangent hyperplane at

Y to the hypersurface with equation XtA Xσ = 0 and for every point Y ∈ \L, Y⊥

is the tangent hyperplane at Y to the hypersurface with equation XtAσ_t−1Xσ−1 = 0.

Both these hypersurfaces have as set ofFqn-rational points the set.

Inspired by the characterization of non-degenerate quadrics, Hermitian varieties and symplectic polar spaces of PG(d, qn) (see Proposition1.12), we define a non-degenerateσ-quadric of PG(d, qn) by induction on the dimension d of the projective space.

Definition 11 Let be a σ-quadric of PG(d, qn), denote by L and R the left and right

radicals of the associated sesquilinear form.

(i)  is a non-degenerate σ-quadric of PG(1, qn) if || ∈ {0, 2, q + 1}. (ii)  is a non-degenerate σ-conic of PG(2, qn), if the following hold:

• L ∩ R = ∅,

• the tangent line L_{to at L intersects  exactly at the point L.}

(iii)  is non-degenerate σ-quadric of PG(d, qn), d ≥ 3, if the following hold: • L ∩ R = ∅,

• ∀ Y ∈ L, the set Y_{∩  is a cone (Y , Q) with vertex Y and base a}

non-degenerate σ-quadric Q in a (d − 2)-dimensional subspace S not through

Y .

2.1 The-quadrics of PG(1, qn) and of PG(2, qn)

In this subsection, we recall what is known for theσ-quadrics of PG(1, qn) and of PG(2, qn) with Fix(σ) = Fq(see [9,14]).

In what follows, we will always assume that σ = 1 since if σ = 1, the set  is either a (possibly degenerate) quadric or a (possibly degenerate) symplectic geometry. Moreover, since Fi x(σ) = Fqwe have thatσ : x → xqm,(m, n) = 1.

Proposition 2.6 Aσ-quadric of PG(1, qn_{) is one of the following:}

• an Fq-subline PG(1, q) of PG(1, qn).

Obviously the following holds:

Proposition 2.7 Let and be aσ-quadric and σ-quadric of PG(1, qn), respec-tively. The sets and are PL-equivalent if and only if || = ||.

The sets of the absolute points ofσ-correlations of PG(2, qn) have been completely determined by the huge work of B.C. Kestenband in 11 different papers from 1990 to 2014. There are lots of different classes of such sets. The interested reader can find all of them in 11 different papers from 1990 to 2014 covering 400 pages of mathematics (see [17–27]). In what follows, we will use the following:

Definition 12 A Kestenbandσ-conic of PG(2, qn) is the set of absolute points of a σ -correlation, σ = 1, of PG(2, qn_).

Proposition 2.8 Aσ-conic of PG(2, qn_{) is one of the following:}

• a cone with vertex a point A projecting a σ -quadric of a line not through A. • a degenerate Cm

F-set(i.e., the union of a line with a scattered linear set iso-morphic to PG(n, q), meeting the line in a maximum scattered Fq-linear set of pseudoregulus type),

• a Cm

F-set(i.e., the union of q −1 scattered Fq-linear sets each of which isomorphic to PG(n − 1, q) with two distinct points ).

• a Kestenband σ-conic

Proposition 2.9 Let be a σ-conic of PG(2, qn) with two different vertices R and L. Associated to there is a σ-collineation between the pencils of lines with vertices R and L. Moreover, is also a σ−1conic.

Proof To the σ-conic , there is associated a σ-collineation  between the pencils of

lines with vertices R and L s.t. is the set of points of intersection of corresponding lines under (resp. ) (see [9]). Moreover, theσ -conic  with associated the σ-collineation is also a σ−1conic with associated theσ−1-collineation−1. 

Proposition 2.10 Let be a σ -conic of PG(2, qn_{) with two different vertices R, L and} letbe aσ-conic of PG(2, qn) with two different vertices R, L. The sets and  are PL-equivalent if and only if either σ= σ or σ= σ−1.

Proof Let σ : x → xqm

and letσ : x → xqm with(m, n) = (m, n) = 1. We may assume that R = R and L = L since the group PL(3, qn) is two-transitive on points. Let (resp. ) be theσ -collineation (resp. σ-collineation) associated with (resp. ). First, assume that is Cm_F-set and thatis a Cm_F-set. Let f be a collineation of PG(2, qn) mapping  into . Since R and L are the unique points of both and not incident with(q + 1)-secant lines, it follows that f stabilizes the set {R, L}. First, assume that f (R) = L. For every line through the point

R, we have that f(( )) = ()−1( f ( )). As  and ()−1are collineations with accompanying automorphismσ and σ−1, it follows thatσ= σ−1. Next suppose that

f(R) = R, f (L) = L. For every line through R, we have that f (( )) = ( f ( )),

and henceσ = σ.

Next assume that is a degenerate Cm_F-set and that is a degenerate Cm_F-set and that n ≥ 3. Let f be a collineation of PG(2, qn) mapping  into . Since the line

R L is the unique line contained in both and , the collineation f stabilizes R L. The directions of the sets\RL and \RL on the line RL are both Fq-linear sets of

pseudoregulus type with transversal points R and L. Since the transversal points of anFq-linear set of pseudoregulus type of PG(1, qn), n ≥ 3 are uniquely determined

(see Proposition 4.3 in [30]), it follows that f stabilizes the set{R, L}. The assertion

follows with the same arguments as in the previous case. 

Finally, the following holds:

Proposition 2.11 Let be a σ-conic of PG(2, qn) with vertices R = L, and let be aσ-conic of PG(2, qn) with vertices R= L. The sets and are PL-equivalent if and only if|| = ||.

Proof We may assume that R = R_{since the group PL(3, q}n_{) is transitive on points.}

Obviously, if and are PL-equivalent, then || = ||. Vice versa if || = ||, then both and are cones with vertex the point R projecting aσ -quadric on a line

not through R. The assertion follows since two σ -quadrics of are PL-equivalent

if and only if have the same size (see [9]). 

Remark 2.12 We can divide the σ-conics in two families:

• the degenerate σ-conics: the degenerate Cm

F-sets and the cones with vertex a point V projecting either the empty set or a point or two points or q+ 1 points on a line not through the point V ;

• the non-degenerate σ-conics: the Kestenband σ -conics and the Cm F-sets.

In the remaining part of this chapter, we will determine the canonical equations and some properties for theσ-quadrics of PG(3, qn) associated to a degenerate correlation of PG(3, qn). Let  be such a σ -quadric with equation XtA Xσ = 0, A a singular

matrix. We will consider three separate cases according to r k(A) ∈ {1, 2, 3}.

3 Seydewitz’s and Steiner’s projective generation of quadrics of

PG

(3, F)

In this section, we recall the projective generation of F. Seydewitz and J. Steiner of quadrics of PG(3, F), F a field. We start with Seydewitz’s construction. In what follows, if P is a point of PG(3, F) we will denote with SPthe star of lines with center

P and withS∗_P the star of planes with center P.

Theorem 3.1 (F. Seydewitz [36]) Let R and L be two distinct points of PG(3, F) and let

 : SR −→ S∗

L be a projectivity. The set of points of intersection of corresponding elements under is one of the following:

• If (RL) is a plane through the line RL, then  is either a quadratic cone or a

• If (RL) is a plane not through the line RL, then  is a empty,

non-degenerate quadric, i.e., either an elliptic quadricQ−(3, F) or an hyperbolic quadricQ+(3, F).

Next consider Steiner’s construction. In what follows, if s is a line of PG(3, F) we will denote byPsthe pencil of planes through s.

Theorem 3.2 (J. Steiner [37]) Let r and be two skew lines of PG(3, F). Let  :

Pr −→ P be a projectivity. The set of points of intersection of corresponding planes under is a hyperbolic quadric Q+(3, F) of PG(3, F).

Proposition 3.3 In PG(2, F) the set of the absolute points of a linear (possibly degen-erate) correlation satisfies an equation XtA X = 0, for some matrix A. Therefore, it is one of the followings:

• the empty set (e.g., F = R) • a point, a line, two lines, • a non-degenerate conic,

• a degenerate symplectic geometry PR, for some point R.

Proof It follows immediately from Proposition1.14.  In Seydewitz’s construction, if we assume that the points R and L coincide, then we get the following:

Proposition 3.4 Let R be a point of PG(3, F). Let  : SR −→ S_R∗ be a projectivity. The set of points of intersection of corresponding elements under  is one of the following:

• the point R (e.g., F = R),

• a line through R, a plane through R, two distinct planes through R,

• a cone with vertex the point R and base a non-empty, non-degenerate conic in a

plane not through R,

• a degenerate symplectic geometry Pr, for some line r through R.

Proof The set of points of intersection of corresponding elements under  is a cone

with vertex the point R projecting the set of absolute points of a linear correlation of a planeπ not through the point R. Hence, the assertion follows from the previous

proposition. 

In Steiner’s construction, if the lines r and either intersect at a point V or coincide, then we get the following:

Proposition 3.5 Let r and be two lines s.t. r ∩ = {V }. Let  : Pr −→ P be a projectivity. The set of points of intersection of corresponding planes under is one of the following:

• a pair of distinct planes,

• a cone with vertex the point V and base a non-empty, non-degenerate conic in a

Proposition 3.6 Let r be a line and let : Pr −→ Pr be a projectivity. The set of points of intersection of corresponding planes under is one of the following:

• the line r, • a plane,

• a pair of distinct planes,

• the degenerate symplectic geometry Pr.

Remark 3.7 If F is either an algebraically closed field or a finite field, then

Seyde-witz’s construction in PG(3, F) gives all the possible quadrics of PG(3, F) and also the degenerate symplectic geometry Pr, for some line r . If F = R is the field of real numbers, then the only missing quadric, up to projectivities, is the quadric with equation x₁2+ x₂2+ x₃2+ x₄2 = 0, that gives as set of points in PG(3, F), the empty set. With Steiner’s construction in PG(3, F), we miss also the elliptic quadric. Note that also the converse holds:

Proposition 3.8 Let be either a, non-empty, quadric or a degenerate symplectic geometry of PG(3, F), F being a field. There exists two point R and L of  and a projectivity : SR −→ S_L∗s.t. is the set of points of intersection of corresponding elements under.

4

-quadrics of rank 3 in PG(3, q

)

Let : XtA Xσ = 0 be a σ -quadric of PG(3, qn) associated to a σ -sesquilinear form

 , . In this section, we assume throughout that rk(A) = 3. Therefore, the radicals

V⊥and Vare one-dimensional vector subspace spaces of V , so they are points of PG(3, qn). We distinguish several cases:

1) V⊥= V.

We may assume w.l.o.g. that the point R = (1, 0, 0, 0) is the right radical and the point L= (0, 0, 0, 1) is the left radical. It follows that

A= ⎛ ⎜ ⎜ ⎝ 0 a12 a13 a14 0 a22 a23 a24 0 a32 a33 a34 0 0 0 0 ⎞ ⎟ ⎟ ⎠ , and  : (a12x1+ a22x2+ a32x3)x2σ+ (a13x1+ a23x2+ a33x3)x3σ +(a14x1+ a24x2+ a34x3)x4σ = 0.

The degenerate collineation

associated to the sesquilinear form maps points into planes through the point L. Points that are on a common line through R are mapped into the same plane through L. Therefore, induces a collineation  : SR −→ S∗_L. Let

SR= { α,β,γ : (α, β, γ ) ∈ PG(2, qn)}, where α,β,γ : ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ x1= λ x2= μα x3= μβ x4= μγ , (λ, μ) ∈ PG(1, qn₎ and S∗L = {πα,β,γ : (α, β, γ) ∈ PG(2, qn)}, where πα,β,γ : αx1+ βx2+ γx3= 0.

The collineation is given by ( _α,β,γ) = π_α_,β_,γ, with

(α_{, β}_{, γ}_{)t = A}_{(α, β, γ )}σ t,

where A is the matrix obtained by A by deleting the last row and the first column. Note that|A| = 0 since rk(A) = 3. It is easy to see that  is the set of points of intersection of corresponding elements under the collineation.

If Y = (y1, y2, y3, y4) is a point of \{R}, then the tangent plane to  at the point Y

is the planeπY = Ywith equation XtAYσ = 0. It follows that for every point Y

of\{R} the plane πY contains the point L. The tangent planeπL = Lto at the point L is the plane with equation XtALσ = 0, that is:

πL : a14x1+ a24x2+ a34x3= 0.

We again distinguish some cases.

• First, assume that πL contains the line R L.

It follows that, w.l.o.g., we may putπL : x3= 0. Hence, a14 = a24 = 0 and we

can put a34 = 1, obtaining

 : (a12x1+ a22x2+ a32x3)x2σ+ (a13x1+ a23x2+ a33x3)x3σ+ x3x4σ = 0.

With this assumption, the collineation  maps the line L R into the plane πL. Consider now the pencilPR,πL of lines through R inπL. We distinguish two cases.

i) maps the lines of PR_,πL into the planes through the line R L.

In this case, we can assume that maps the line x3= x4= 0 into the plane x2= 0

and the line x2= x4= 0 into the plane x1= 0 obtaining  : ax1x3σ + bx2σ+1+ x3x4σ = 0.

We can assume that contains the points (0, 1, −1, 1) and (1, 0, 1 − 1) obtaining

a= b = 1 and hence a canonical equation is given by  : x1x3σ + x2σ+1+ x3x4σ = 0.

Note that in this case is the set of points studied in [13], where has been called a

σ-cone. In this paper, we call this set a degenerate parabolic σ-quadric with collinear vertex points R and L.

In [13] it has been proved that the following holds:

Theorem 4.1 Let be a degenerate parabolic σ-quadric  of PG(3, qn) with collinear vertex points R and L. Then|| = q2n+ qn+ 1, RL is the unique line contained in  and πLis the unique plane that meets exactly in RL.

ii) does not map the lines of PR_,πL into the planes through the line R L. In this case, there exists a planeπ containing RL such that the lines of the pencil

PR,πare mapped, under into the planes through RL. Hence, there is a unique line through R (beside R L) contained in. In this case, we may assume that  maps the line x3 = x4 = 0 into the plane x1 = 0 and the line x2 = x4 = 0 into the plane x2= 0. Hence:

 : ax1x2σ + bx2x3σ + x3x4σ = 0.

Assuming that contains the points (0, 1, 1, −1) and (1, 1, −1, 0), we get a = b = 1 and hence a canonical equation is given by

 : x1x2σ + x2x3σ + x3x4σ = 0.

We will call this set a hyperbolicσ -quadric with collinear vertex points R and L.

Theorem 4.2 Let  be a hyperbolic σ-quadric of PG(3, qn) with collinear vertex points R and L. Then|| = (qn+ 1)2and contains exactly three lines: the line R L, a unique other line through R and a unique other line through L.

• πLdoes not contain the line R L (or equivalently does not map the line RL into

a plane through the line R L).

W.l.o.g. we may putπL : x1 = 0. In this case, there is a plane through R (not

containing L), sayπR, such that the pencil of lines through R inπR is mapped, under, into the pencil of planes through RL. We may assume that πR : x4= 0.

Hence, maps the lines _α,β,0into the planesπ0,β,γ, so we may assume that

maps the line 1,0,0into the planeπ0,1,0and the line 0,1,0into the planeπ0,0,1.

Hence, the points of satisfy the equation

ax₂σ+1− bx₃σ+1+ x1x4σ = 0.

Assuming, w.l.o.g., that the point(1, 1, 0, −1) belongs to , we obtain a = 1. The number of lines through R contained in depends on the number of solutions of the

equation xσ+1= b, and hence, it is either 0, 1, 2 or q + 1 depending upon q even or odd and n even or odd. We distinguish several cases:

– If q is even and n is even, then there are either 0 or 1 or q+ 1 solutions giving either 0 or 1 or q+ 1 lines through R (and hence through L) contained in . – If q is even and n is odd, then there is a unique solution of the equation giving one

line through R and one line through L contained in.

– If q is odd and n is even, then there are either 0 or q+ 1 solutions of the equation giving either 0 or q+ 1 lines through R (and through L) contained in .

– If q is odd and n is odd, then there are either 0 or 2 solutions of the equation giving either 0 or 2 lines through R (and through L) contained in.

In these cases, we will call the set either an elliptic or a parabolic or a hyperbolic or a(q + 1)-hyperbolic σ -quadric with vertex points R and L according to the number of lines through R contained in is either 0 or 1 or 2 or q + 1.

If q is even and n is even, put r = (qn− 1, qm+ 1).

Theorem 4.3 If is an elliptic σ -quadric of PG(3, qn_{) with vertex points R and L,} then has canonical equation x₂σ+1− bxσ+1₃ + x1x₄σ = 0, with b a non-square if q is odd and b(qn−1)/r = 1 if q is even and n is even. Moreover, || = q2n + 1 and  contains no line.

Theorem 4.4 If is a parabolic σ-quadric of PG(3, qn_{) with vertex points R and L,} then q is even and has canonical equation x₂σ+1− bx₃σ+1+ x1x₄σ = 0, where the equation xσ+1 = b has a unique solution. Moreover, || = q2n _{+ q}n _{+ 1 and } contains a unique line through R and a unique line through L.

Theorem 4.5 If is a hyperbolic σ -quadric of PG(3, qn_{) with vertex points R and L,} then both q and n are odd and has canonical equation x₂σ+1− bx₃σ+1+ x1x₄σ = 0, where xσ+1 = b has exactly two solutions. Moreover, || = q2n + 2qn+ 1 and  contains exactly two lines through R and exactly two lines through L.

Theorem 4.6 If is a (q + 1)-hyperbolic σ -quadric of PG(3, qn_{) with vertex points} R and L, then n is even and has canonical equation x₂σ+1− bx₃σ+1+ x1x₄σ = 0, where xσ+1 = b has exactly q + 1 solutions. Moreover, || = q2n + (q + 1)qn+ 1 and contains exactly q + 1 lines through R and exactly q + 1 lines through L.

2) V⊥= V.

We may assume w.l.o.g. that the point R= L = (1, 0, 0, 0) is both the left radical and the right radical. It follows that, in this case, is a cone with vertex the point R. Since the matrix A has rank three with first column and last row equal to 0, by choosing a plane not through the point R, e.g.,π : x1= 0, we get that the set  ∩ π is a σ-conic

of the planeπ with associated matrix of rank 3. Hence, it is a Kestenband σ-conic of

π. It follows that  is a cone with vertex the point R projecting a Kestenband σ-conic

5

-quadrics of rank at most 2 in PG(3, q

)

In this section, a σ-quadric  of PG(3, qn) will have equation XtA Xσ = 0 with r k(A) ≤ 2.

5.1-quadrics of rank 2

Since r k(A) = 2, it follows that the left and right radicals are two lines r and of PG(3, qn). We distinguish three cases.

1) r∩ = ∅.

We may assume w.l.o.g. that r: x3= x4= 0 and : x1= x2= 0. Then:  : (a31x3+ a23x2)x3σ + (a14x1+ a24x2)x4σ = 0,

that is the set of points of PG(3, qn) of intersection of corresponding planes under a collineation : Pr −→ P , where

Pr = {πa,b: ax3+ bx4= 0}{(a,b)∈PG(1,qn_)},

P = {πa,b: ax1+ bx2= 0}{(a,b)∈PG(1,qn_)}.

The set contains the qn+1 lines of a pseudoregulus (see [30]) with transversal lines

r and We call this set a σ-quadric of pseudoregulus type with skew vertex lines r and .

Note that if n= 2, σ2= 1 a pseudoregulus have been already introduced in [15] and

σ-quadrics of pseudoregulus type with skew vertex lines have been introduced in [11] where they were called hyperbolic QF-sets.

Let : Pr −→ Psbe a collineation with accompanying automorphismσ : x → xq

m ,

(m, n) = 1.

Assuming that :

(π1,0) = π1_,0, (π0,1) = π0_,1, (π1,1) = π1_,1,

we have that(πa_,b) = π_aσ_,bσ.

Hence, the setQ of points of intersection of corresponding planes under  is given by the points whose homogeneous coordinates are the solutions of the linear system

ax3+ bx4= 0 aσx1+ bσx2= 0,

where(a, b) ∈ PG(1, qn_{). This system is equivalent to the linear system}

ax3+ bx4= 0 ax₁σ−1 + bx₂σ−1 = 0.

A point P= (x1, x2, x3, x4) belongs to Q if and only if the previous linear system, in

the unknowns a and b, has non-trivial solutions, hence if and only if x₁σ−1x4−x₂σ−1x3=

0, that is x1x₄σ − x2x₃σ = 0. This can be seen as a canonical equation of a σ-quadric

of pseudoregulus type with skew vertex lines. Let c∈ F∗qand letγ ∈ F∗qn be such that

N(γ ) = c. Put

Qc= {(γ xσ, γ yσ, x, y)}(x,y)∈PG(1,qn₎.

The setQcis a maximum scattered linear set of rank 2n of PG(3, qn_{) of pseudoregulus}

type with transversal lines r and . Hence, the set Q is the union of the skew lines r,

and the q − 1 linear sets of pseudoregulus type Qc, c∈ F∗_qn. The following holds:

Proposition 5.1 Let be a σ-quadric of pseudoregulus type of PG(3, qn) with skew transversal lines r and . The only lines contained in  are r, and the qn+ 1 lines of the pseudoregulus associated to.

Proposition 5.2 Let a, b and c be three pairwise skew lines and let and s be two skew lines meeting a, b and c. There is a unique σ -quadric of pseudoregulus type of

PG(3, qn) with skew vertex lines r and containing a, b and c. 2) r∩ = {V } is a point.

We may assume w.l.o.g. that r : x3 = x4 = 0, : x2 = x3 = 0. In this case, the σ -quadric  is a cone with vertex the point V projecting a (degenerate or not) Cm

F-set

in a plane not through V . Indeed, let π be a plane not through the point V and let

R = r ∩ π, L = ∩ π. It follows that  ∩ π is a set of points of π generated by a

collineation between the pencils of lines ofπ with center the points R and L induced by the collineation between the pencil of planesPr andP that is associated to. 3) r= .

We may assume w.l.o.g. that r = : x3 = x4 = 0. In this case, the σ -quadric is a

cone with vertex the line r over aσ -quadric of a line skew with r. That is  is either just the line r or a plane through r or a pair of distinct planes through r or q+ 1 planes through r forming anFq-subpencil of planes through r .

5.2-quadrics of rank 1

In this subsection, aσ-quadric  will have equation XtA Xσ = 0 with rk(A) = 1.

Hence, dimV⊥= dim V= 3 so left and right radicals in PG(3, qn) are planes. We distinguish two cases:

• V⊥_{= V}_.

We may assume that r : x4 = 0 is the right radical and : x1 = 0 is the left

radical. Hence, : x1x4σ = 0, that is the union of two different planes.

• V⊥_{= V}_.

We may assume r = : x4 = 0 is both the left and right radical. Hence,  : x₄σ+1= 0, that it is a plane of PG(3, qn).

6

-quadrics of PG(3, q

) with |A| = 0

In this section, we summarize the results obtained onσ -quadrics.

Proposition 6.1 Let : XtA Xσ = 0 be a σ-quadric of PG(3, qn), with|A| = 0. The set is one of the following:

• a cone with vertex a line v projecting a σ -quadric of a line skew with v (hence

either just the linev or one, two or q + 1 planes through v),

• a cone with vertex a point V projecting either a Kestenband σ -conic or a (possibly

degenerate) Cm_F-set of a planeπ, with V /∈ π,

• a degenerate either parabolic or hyperbolic σ-quadric with two collinear vertex

points,

• a non-degenerate either elliptic or parabolic or hyperbolic or (q + 1)-hyperbolic

σ-quadric with two vertex points,

• a non-degenerate σ -quadric with two skew vertex lines (i.e., a σ -quadric of

pseu-doregulus type).

Letσ : x → xqm,σ: x → xqm,(m, n) = (m, n) = 1.

Proposition 6.2 Let be a σ -quadric and let be aσ-quadric of PG(3, qn). If  andare cones, then and are PL-equivalent if and only if the basis of the cones are PL-equivalent.

We will say that aσ-quadric and a σ-quadric of PG(3, qn) are of the same type if they have the same name.

Proposition 6.3 Let be a σ-quadric and let be aσ-quadric of PG(3, qn_{). If } andare not cones and have non-empty vertices, then and are PL-equivalent if and only if eitherσ= σ or σ= σ−1and, are of the same type.

Proof Since  and  _{are not cones, both and } _{have two distinct vertices. We}

divide two cases.

Case 1. The vertices of both and are points.

We may assume that and have the same vertices R and L. Let (resp. ) be theσ -collineation (resp. σ-collineation) between the star of lines with center R and the star of planes with center L associated to (resp. ). Observe that is also a

σ−1_{-quadric with vertices L and R to which there is associated theσ}−1_{-collineation} −1_.

Let f be a collineation of PG(3, qn) mapping  into . Since the number of lines of

 through either R or L is at most q + 1, there exists plane π through the line RL

s.t. the lines ofπ through R are mapped under  onto planes through a line on L not contained inπ. It follows that  induces a collineation π between the pencil of lines through R and the pencil of lines through L. This gives that ∩ π is a (possibly degenerate) Cm_F-set ofπ. It follows that f ( ∩ π) is a (possibly degenerate) either

Cm_F or Cn_F−m-set. Hence,σ= σ or σ= σ−1.

Case 2. The vertices of bothand are two skew lines.

We may assume that and have the same vertices r and . Every plane not through

r neither through meets  in a (possibly degenerate) Cm_F set. The assertion follows

7

-quadrics of rank 4 of PG(d, q

)

Opposite to the case of PG(2, qn), nothing is known for the sets of absolute points of PG(d, qn) of a nonlinear correlation, different from a polarity, of PG(d, qn), d ≥ 3. We recall that took roughly 14 years and 10 different papers to B. Kestenband to classify allσ -conics of PG(2, qn) with equation XtA Xσ = 0, |A| = 0. In this section,  will

be aσ -quadric of PG(d, qn) with equation XtA Xσ = 0, |A| = 0. We determine the

dimension of the maximum totally isotropic subspaces contained in.

Proposition 7.1 If Sh is an h-dimensional subspace of PG(d, qn) contained in a σ-quadric with equation XtA Xσ = 0, |A| = 0, then h ≤ d−1₂ . Moreover, there exists aσ -quadric with equation XtA Xσ = 0, |A| = 0, containing a subspace with dimension d−1₂ .

Proof Let S be a subspace with maximum dimension, say h, contained in . We may

assume, w.l.o.g., that S has equations xh+2= xh+3= . . . xd+1= 0. It follows that the

matrix A has the submatrix formed by the first h+1 rows and h+1 columns with entries all 0’s. This gives that d+1 = rank(A) ≤ 2(d −h), and hence h ≤ d−1₂ . If d is odd, then theσ -quadric with equation x1xd+1σ+ x2xdσ+ . . . + xd+1x1σ = 0 contains the

d−1

2 -dimensional subspace with equations x1= x2= . . . = xd+1₂ = 0. If d is even,

then theσ -quadric with equation x1xd₊₁σ+ x2xdσ+. . .+ xσ+1d

2+1

+. . .+ xd₊₁x1σ = 0

contains the(d₂−1)-dimensional subspace with equations x1= x2= . . . = xd

2+1= 0.

 A similar proof can be given to obtain the following, more general, result

Proposition 7.2 If Sh is an h-dimensional subspace of PG(d, qn) contained in a σ-quadric  with equation XtA Xσ = 0, then h ≤



d−rank₂(A)



. Moreover, there exists aσ -quadric with equation XtA Xσ = 0, containing a subspace with dimension



d−rank₂(A)



.

8 Applications of

-quadrics of PG(3, 2

)

8.1-quadrics of pseudoregulus type and the Segre’s (2n+ 1)-arc of PG(3, 2n),

An arc in PG(3, q) is a set of points with the property that any 4 of them span the whole space. The maximum size of an arc in PG(3, q) is q + 1 as proved by Segre in [34] for q odd and by Casse in [6] for q even. An example of(q + 1)-arc of PG(3, q),

q = 2n, was given by Segre in [35] and it is a set of points projectively equivalent to the set{(1, t, tσ, tσ+1)} ∪ {(0, 0, 0, 1)}, where σ : x → x2m,(m, n) = 1. The

(q + 1)-arcs of PG(3, q) have been classified by Segre for q odd, q ≥ 5 [34] and by Casse and Glynn for q even, q ≥ 8 [7] and are the twisted cubic and, for q even, the Segre’s(q + 1)-arc.

An ovoid of Q+(3, q) is a set of q + 1 points no two collinear on the quadric. A

of Q+(3, q) fixing P linewise and acting sharply transitively on the points of O\{P}. An example of translation ovoid of Q+(3, q) is a non degenerate conic.

Proposition 8.1 Let be a σ-quadric of pseudoregulus type of PG(3, qn) with skew vertex lines r and . The set  is mapped, via the Klein correspondence, into the union of a translation ovoid of a hyperbolic quadricQ+(3, qn) with two points R and S. Moreover, if q= 2, then this translation ovoid is the Segre’s (2n+1)-arc of PG(3, 2n). Proof Let r : x3 = x4 = 0 and : x1 = x2 = 0 and let  be the σ -quadric with

vertex lines and r with equation x1x4σ − x2x3σ = 0. The lines of the associated

pseudoregulus are spanned by the points(0, 0, α, β) and (cασ, cβσ, 0, 0), (α, β) ∈ PG(1, qn), c ∈ F_q∗. Via the Klein correspondence, the lines of the pseudoregulus are mapped to the set of points {(0, ασ+1, ασβ, αβσ, βσ+1, 0)}_{{(α,β)∈PG(1,q}n_)}, i.e., the set of points of the ovoid

O = {(0, ασ+1, ασ, α, 1, 0)} ∪ {P∞= (0, 0, 0, 0, 1, 0)}

of the hyperbolic quadricQ+(3, qn) with equations x1= x6 = 0, x2x5− x3x4 = 0

contained in the Klein quadric with equation x1x6− x2x5+ x3x4 = 0. The two

vertex lines r and are mapped, via the Klein correspondence, into the two points

(0, 0, 0, 0, 0, 1) and (1, 0, 0, 0, 0, 0) on the line x2 = x3 = x4 = x5 = 0 that is the

polar line w.r.t. the three-dimensional subspace with equations x1= x6= 0 containing

the ovoidO under the polarity defined by the Klein quadric.

The ovoid O is a translation ovoid w.r.t. the point P_∞ of the hyperbolic quadric

Q+_{(3, q}n_{) since the group of projectivities of PG(5, q}n_{) induced by the matrices}

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 0 0 0 0 0 0 1 0 0 0 0 0 b 1 0 0 0 0 bσ 0 1 0 0 0 bσ+1bσ b 1 0 0 0 0 0 0 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

b∈ Fqn, stabilizes P_∞and acts sharply transitively on the points ofO\{P_∞}. 

8.2 Degenerate-quadrics and Lüneburg spread of PG(3, 2n).

In 1965, H. Lüneburg proved that if qn= 22h+1, h≥ 1, then the set of absolute lines of a polarity of W(3, qn) is a symplectic spread, now called the Lüneburg spread of PG(3, qn) (see [32]).

Let _∞be a hyperplane of PG(4, qn) and let Q+(3, qn) be a hyperbolic quadric of

∞. A setA of q2n points of PG(4, qn)\ ∞s.t. the line joining any two of them is disjoint fromQ+(3, qn_{) is called an affine set of PG(4, q}n_)\

∞. In what follows, we will denote by⊥ the polarity of Q+(5, qn_{). In [29] and also in [31] the following}

Theorem 8.2 LetO be an ovoid of Q+(5, qn), let P be a point of O and let  be a hyperplane of PG(5, qn) not containing P. The set AP(O) obtained by projecting O from the point P into is an affine set of \P⊥. Conversely, ifA is an affine set of \P⊥_{, then the setO = {P R ∩ Q}+_{(5, q}n_{) : R ∈ A} is an ovoid of Q}+_{(5, q}n_).

IfS is a spread of PG(3, qn) and is a line of S, then we will denote by A (S) the affine set arising fromS with respect to .

If S is a symplectic spread, then A (S) is a set of q2n points of an affine space PG(3, qn)\π_∞ such that the line joining any two of them is disjoint from a given non-degenerate conicC of π∞.

The affine set arising from the Lüneburg spread has been studied by A. Cossidente, G. Marino and O. Polverino in [8], where the following result has been obtained:

Theorem 8.3 The affine setA of the Lüneburg spread of PG(3, 22h+1) is the union of 22h+1-arcs, and each of them can be completed to a translation hyperoval. The directions ofA on π_∞are the complement of a regular hyperoval.

With the following theorem, we observe that the affine set of a Lüneburg spread of PG(3, 22h+1_{) is the affine part of a degenerate elliptic σ -quadric of PG(3, 2}2h+1_{) with}

two vertex points.

Theorem 8.4 Let be a degenerate parabolic σ-quadric of PG(3, 2n) with collinear vertex points R and L. The lines joining any two points of \RL are disjoint from a translation hyperovalO_∞of the planeπL, projectively equivalent to the set

{(0, t, tσ−2

, 1) : t ∈ F2n} ∪ {R, L}.

If n= 2h + 1 and σ is the automorphism of F2n given byσ : x → x2 h

, thenO_∞is a regular hyperoval. Hence, the set\RL is the affine set of the Lüneburg spread of

PG(3, qn).

Proof Let  be a degenerate parabolic σ-quadric with collinear vertex points R = (0, 0, 1, 0) and L = (0, 1, 0, 0) with equation

x4σ+1+ x1x2σ − x3x1σ = 0.

It follows that the planeπR L has equation x1 = 0. The set A = K\πR L is given by A = {(1, x, xσ _{+ y}σ+1_{, y) : x, y ∈ Fq}n}. Arguing as in Proposition 5.2 in [8], we obtain that the set of directions determined byA into the plane πR Lcovers all the points ofπR Lexcept to the points of a hyperovalC given by C = {(0, x, xσ−2, 1) : x ∈ F∗_qn}. Note that if q = 22h+1andσ : x → x2h, then the hyperovalC is a hyperconic. The

assertion follows (see Sect.8.2). 

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CRUI-CARE Agreement.

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